The Future World : A train which runs inside Earth's core

A gravity train is a
theoretical means of transportation intended to go between two points on the
surface of a sphere, following a straight tunnel that goes directly from one
point to the other through the interior of the sphere.

In a large body such as a
planet, this train could be left to accelerate using just the force of gravity,
since, during the first half of the trip (from the point of departure until the
middle), the downwards pull towards the center of gravity would pull it towards
the destination. During the second half of the trip, the acceleration would be
in the opposite direction relative to the trajectory, but (ignoring the effects
of friction) the speed acquired before would be enough to cancel this
deceleration exactly (so that the train would reach its destination with speed
equal to zero).

In reality, there are two
reasons gravity trains do not exist. First, the transit shown in the
illustration would pierce the Earth's mantle and traverse a region where rock
is more fluid than solid. No materials are known that would withstand the
tremendous heat and pressure in the inner core. Temperature is estimated as
5,700 K (5,430 °C; 9,800 °F), and pressure as high as about 330 to 360
gigapascals (3,300,000 to 3,600,000 atm). Secondly, frictional losses would be
significant. Rolling friction losses could be reduced by using a magnetically
levitated train. However, unless all air is evacuated from the tunnel,
frictional losses due to air resistance would render the gravity train
unusable. Evacuating the atmosphere to make it a vactrain would eliminate this
drag but would require additional power. Such objections would not apply for
solid planets and moons that do not have an atmosphere.

Where does this concept origin ?

In the 17th century, British
scientist Robert Hooke presented the idea of an object accelerating inside a
planet in a letter to Isaac Newton. A gravity train project was seriously
presented to the Paris Academy of Sciences in the 19th century. The same idea
was proposed, without calculation, by Lewis Carroll in 1893 in Sylvie and Bruno
Concluded. The idea was rediscovered in the 1960s when physicist Paul Cooper
published a paper in the American Journal of Physics suggesting that gravity
trains be considered for a future transportation project.

What are the mathematical considerations behind it ?

The gravity train has
several curious properties.All straight-line gravity
trains on a given planet take exactly the same amount of time to complete a
journey (that is, no matter where on the surface the two endpoints of its
trajectory are located). For Earth, this time would equal 2530.30 seconds
(nearly 42.2 minutes) if it were a perfect sphere and of uniform density.

The time of a trip depends
only on the density of the planet and the gravitational constant (except when
travelling at a significant proportion of the speed of light).

The maximum speed is reached
at the middle point of the trajectory. For a train that goes directly through
the center of the Earth, this maximum speed is about 7,900 metres per second
(28440 km/h).The Time period of this
simple harmonic motion is 84 minutes approximately.The derivation assumes that
the mass is distributed homogeneously throughout the earth. The shortest time
tunnel through a homogeneous earth is a Hypocycloid.